126 XIII INTEGRATION Since the sequence (gn) is increasing and \ gn d[i^ I * gn dpi < + oo, the result follows from (13.8.1). (13.8.3) Let (fn) be a sequence of integr able functions, and suppose that there /* hdjn< 4-00 and — h ^fn :g h almost every where, for alln. Then the functions lim inf/, andlim sup/, are integrable, «-+ oo n~*oo and we have (13.8.3.1) Aiminf/B) d\i £ lim inf /„ d/j, J \ n~*oo / n-*oo J ^ lim sup Ifndfj,^ (lim sup/J d\i. n~*oo J J \ n-*-oo / It is enough to prove that lim inffn is integrable and that the left-hand n-»-oo inequality in (13.8.3.1) is valid; the other assertions then follow by replacing fn by — /, . Let gn = inf/n+ p . It follows from (1 3.8.2) applied to the sequence p^O (—fn+p)P£o and from ^e hypothesis fn+p ^ — h, that gn is integrable, and clearly f gn d/j, £ (fn+p d\i for all p ^ 0, so that f gn d^ g inf (fn+p d\i. The J J J p£QJ sequence (gn) is increasing ; we have gn^h for all n, and by definition (1 2.7.1 0), lim infyj, = sup gn . Hence it follows from (13.8.1) that lim inf/w is integrable n-»oo n n-*oo and that (lim inf/J d/j, = sup gn dp ^ sup ( inf fn+p d/A J \ n~*oo / n J n \p£0 J / = lim inf /„ dfi. w-»-oo J (13.8.4) (Dominated convergence theorem) Let (fn) be a sequence of integrable functions ; suppose that Iimfn(x) =f(x) exists almost everywhere n-* oo and that there exists a function g ^ 0 such that j g d/j, < +00 and \fn\ ^ g almost everywhere, for alln. Then f is integrable and (13.8.4.1) f/(x) dftx) .= lim (/„(*) d/i(x). J B-+00-J This follows from (13.8.3). In this situation the two extreme terms of (13.8.3.1) are equal, whence the result.f:X-+Htobe integrable it is necessary and